How do you simulate the reverse stochastic differential equation (SDE)?

To simulate a reverse stochastic differential equation (SDE), you need to construct a numerical approximation of the time-reversed version of a forward SDE. A reverse SDE models a stochastic process backward in time, often used in applications like diffusion-based generative models or solving inverse problems. The key idea is to derive the reverse-time dynamics by adjusting the drift term to account for the time reversal, which typically involves the score function (gradient of the log-probability density) of the forward process. This adjustment ensures the reverse process aligns with the statistical properties of the original forward SDE when run backward.

Simulating a reverse SDE involves three main steps. First, discretize the continuous-time equation using a numerical scheme like Euler-Maruyama, which approximates the solution at discrete time steps. For example, consider a forward SDE ( dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t ). The reverse SDE (under certain conditions) is ( dX_t = [\mu(X_t, T-t) - \sigma^2(X_t, T-t)\nabla \log p_t(X_t)]dt + \sigma(X_t, T-t)d\widetilde{W}_t ), where ( \nabla \log p_t ) is the score function and ( \widetilde{W}_t ) is a reverse-time Brownian motion. To simulate this, you start from a terminal condition (e.g., noise) and iterate backward, updating ( X_t ) at each step using the adjusted drift and noise. Libraries like PyTorch or JAX can compute gradients for the score function if it’s approximated by a neural network, as in diffusion models.

Practical challenges include accurately estimating the score function and handling the time reversal. For example, in diffusion models, the score is learned via training on data, and the reverse SDE is simulated by discretizing time into small intervals (e.g., 1,000 steps) and applying the Euler-Maruyama update rule backward. Code might involve a loop that decrements time, computes the adjusted drift using the learned score, and adds scaled noise. Stability depends on step size and numerical precision. Tools like SDE solvers in TensorFlow Probability or custom implementations with autograd frameworks simplify this process, but care is needed to ensure the reverse process correctly inverts the forward dynamics.